Unconditional correctness of recent quantum algorithms for factoring and computing discrete logarithms

• URL: http://arxiv.org/abs/2404.16450v1
• Date: Thu, 25 Apr 2024 09:30:19 GMT
• Title: Unconditional correctness of recent quantum algorithms for factoring and computing discrete logarithms
• Authors: Cédric Pilatte,
• Abstract summary: In 2023, Regev proposed a multi-dimensional version of Shor's algorithm that requires far fewer quantum gates. We prove a version of this conjecture using tools from analytic number theory. As a result, we obtain an unconditional proof of correctness of this improved quantum algorithm.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In 1994, Shor introduced his famous quantum algorithm to factor integers and compute discrete logarithms in polynomial time. In 2023, Regev proposed a multi-dimensional version of Shor's algorithm that requires far fewer quantum gates. His algorithm relies on a number-theoretic conjecture on the elements in $(\mathbb{Z}/N\mathbb{Z})^{\times}$ that can be written as short products of very small prime numbers. We prove a version of this conjecture using tools from analytic number theory such as zero-density estimates. As a result, we obtain an unconditional proof of correctness of this improved quantum algorithm and of subsequent variants.

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